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136                        ANALYTICAL MECHANICS
When equation (4) is plotted with, the time as abscissa and the displacement as ordinate the well-known sine curve is obtained, Fig. 71.
It is evident both from equation (4) and from the curve that the maximum value of x is equal to a. This value of the displacement is called the amplitude. The minimum value of # is a displacement equal to the amplitude in the negative direction. Therefore the particle oscillates between the points A and A1'. The displacement equals the positive value of the amplitude every time sin co£ equals unity, that
is, when <*t assumes the values ~, -~, ~, etc.    In other
2i      Z        Z
words, the particle occupies the extreme point A at the instants when t has the values-—-, —, ~-^> etc. Therefore
2 co    2 co    2 co
the particle returns to the same point after a lapse of time •equal to —. This interval of time is called the period of
.the motion and is denoted by P.   Thus
r-~         <«
1.  A particle which moves in a straight groove is acted upon by a force •which is directed towards a fixed point outside the groove, and which varies as the distance of the particle from the fixed point.   Show that the motion is harmonic.
2.   Within the earth the gravitational attraction varies as the distance from the center.   Find the greatest value of the velocity which a .body would attain in falling into a hole, the bottom of which is at the •center of the earth.
3.   Show that when a particle describes a uniform circular motion, its projection upon a diameter describes a harmonic motion.
1. The speed of a train which moves with constant acceleration is doubled in a distance of 3 kilometers. It travels the next I fa kilometers in one minute. Find the acceleration and the initial velocity.